Generalised intermediate dimensions
Generalised intermediate dimensions
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广义中间尺寸
DOI：
10.1007/s00605-023-01884-5
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发表时间：
2020
期刊：
影响因子：
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通讯作者：
Amlan Banaji
中科院分区：
文献类型：
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作者：
Amlan Banaji
我们介绍了一个家庭的尺寸，我们称之为中间尺寸，这之间的Hausdorff和盒尺寸和概括的中间尺寸介绍了法尔科纳，弗雷泽和肯普顿。这是通过限制覆盖集的相对大小来实现的，其方式允许比中间维度的定义更大的细化。我们还扩展了理论从欧氏空间到更广泛的一类度量空间。我们证明，这些尺寸可以用来恢复的Hausdorff和盒尺寸的紧凑子集的中间尺寸是不连续的，从而提供更精细的几何信息，这样的集合之间的插值。我们证明了连续性的结果，涉及Assouad和较低的维度，这给了一个尖锐的一般下界的中间尺寸是积极的allfor集正盒维数。我们还证明了Hölder失真估计，质量分布原理，和Frostman型引理，我们用它来研究产品集的尺寸。
We introduce a family of dimensions, which we call the-intermediate dimensions, that lie between the Hausdorff and box dimensions and generalise the intermediate dimensions introduced by Falconer, Fraser and Kempton. This is done by restricting the relative sizes of the covering sets in a way that allows for greater refinement than in the definition of the intermediate dimensions. We also extend the theory from Euclidean space to a wider class of metric spaces. We prove that these dimensions can be used to ‘recover the interpolation’ between the Hausdorff and box dimensions of compact subsets for which the intermediate dimensions are discontinuous at, thus providing finer geometric information about such sets. We prove continuity-like results involving the Assouad and lower dimensions, which give a sharp general lower bound for the intermediate dimensions that is positive for allfor sets with positive box dimension. We also prove Hölder distortion estimates, a mass distribution principle, and a Frostman type lemma, which we use to study dimensions of product sets.
DOI：
10.2307/2532125
发表时间：
1990-03
期刊：
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影响因子：
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作者：
K. Falconer
通讯作者：
K. Falconer